A client can trust a deeper certificate, but it entirely automatic.
With most TLS servers you have a way to provide a bundle of intermediate certificates in addition to the certificate used for that endpoint.
So the Letsencrypt root is trusted by most browsers/clients
But the certificates from Letsencrypt come from a deeper CA that isn't typically trusted directly. The server must include the intermediate.
So when the client gets connects it will get the intermediates, and the certificate for the resource it is connecting too. It will attempt to trust chain from the root, through the intermediates down to the cert.
So lets look at the case of https://serverfault.com/. My system only trusts the top 'DigiCert' CA, it does not trust 'DigiCert High Assurance EV Root CA'. But I can use https://serverfault.com/ because the server returns the intermediates.
If you want to fire up the CLI you can see the certs with openssl
echo '' | openssl s_client -showcerts -connect serverfault.com:443
depth=2 C = US, O = DigiCert Inc, OU = www.digicert.com, CN = DigiCert High Assurance EV Root CA
depth=1 C = US, O = DigiCert Inc, OU = www.digicert.com, CN = DigiCert SHA2 High Assurance Server CA
depth=0 C = US, ST = NY, L = New York, O = "Stack Exchange, Inc.", CN = *.stackexchange.com
0 s:/C=US/ST=NY/L=New York/O=Stack Exchange, Inc./CN=*.stackexchange.com
i:/C=US/O=DigiCert Inc/OU=www.digicert.com/CN=DigiCert SHA2 High Assurance Server CA
MIIIP... # the intermediate
1 s:/C=US/O=DigiCert Inc/OU=www.digicert.com/CN=DigiCert SHA2 High Assurance Server CA
i:/C=US/O=DigiCert Inc/OU=www.digicert.com/CN=DigiCert High Assurance EV Root CA
MIIEs... # cert for stackexchange
subject=/C=US/ST=NY/L=New York/O=Stack Exchange, Inc./CN=*.stackexchange.com
issuer=/C=US/O=DigiCert Inc/OU=www.digicert.com/CN=DigiCert SHA2 High Assurance Server CA
No client certificate CA names sent
Peer signing digest: SHA512
Server Temp Key: X25519, 253 bits
SSL handshake has read 3945 bytes and written 269 bytes
New, TLSv1.2, Cipher is ECDHE-RSA-AES128-GCM-SHA256
So a client doesn't automatically trust deeper CAs, but it is easy to configure things so certs can be trusted.